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Resolución de integrales/ cqrt(x^2-a^2)

Integral de cqrt(x^2-a^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
  /                
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 |    /  2    2    
 |  \/  x  - a   dx
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0
∫01−a2+x2 dx\int\limits_{0}^{1} \sqrt{- a^{2} + x^{2}}\, dx 
Integral(sqrt(x^2 - a^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
                         //   2      /x\                                                        \
                         ||  a *acosh|-|             3                                  | 2|    |
                         ||          \a/            x                   a*x             |x |    |
                         ||- ----------- + ------------------- - -----------------  for |--| > 1|
                         ||       2                  _________           _________      | 2|    |
  /                      ||                         /       2           /       2       |a |    |
 |                       ||                        /       x           /       x                |
 |    _________          ||                2*a*   /   -1 + --    2*   /   -1 + --               |
 |   /  2    2           ||                      /          2        /          2               |
 | \/  x  - a   dx = C + |<                    \/          a       \/          a                |
 |                       ||                                                                     |
/                        ||                                     ________                        |
                         ||                                    /      2                         |
                         ||                                   /      x                          |
                         ||             2     /x\   I*a*x*   /   1 - --                         |
                         ||          I*a *asin|-|           /         2                         |
                         ||                   \a/         \/         a                          |
                         ||          ------------ + --------------------             otherwise  |
                         \\               2                  2                                  /
∫−a2+x2 dx=C+{−a2acosh⁡(xa)2−ax2−1+x2a2+x32a−1+x2a2for ∣x2a2∣>1ia2asin⁡(xa)2+iax1−x2a22otherwise\int \sqrt{- a^{2} + x^{2}}\, dx = C + \begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{x}{a} \right)}}{2} - \frac{a x}{2 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{x^{3}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{x}{a} \right)}}{2} + \frac{i a x \sqrt{1 - \frac{x^{2}}{a^{2}}}}{2} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
  1                                                                                                 
  /                                                                                                 
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 |  /                            2                   4                     2               2        
 |  |         a                 x                   x                   3*x               x         
 |  |- --------------- + ---------------- - ----------------- + -------------------  for ---- > 1   
 |  |        _________                3/2                 3/2             _________      | 2|       
 |  |       /       2        /      2\           /      2\               /       2       |a |       
 |  |      /       x         |     x |         3 |     x |              /       x                   
 |  |     /   -1 + --    2*a*|-1 + --|      2*a *|-1 + --|      2*a*   /   -1 + --                  
 |  |    /          2        |      2|           |      2|            /          2                  
 |  |  \/          a         \     a /           \     a /          \/          a                   
 |  |                                                                                               
 |  |                    ________                                                                   
 |  <                   /      2                                                                  dx
 |  |                  /      x                                                                     
 |  |          I*a*   /   1 - --                                                                    
 |  |                /         2                                 2                                  
 |  |              \/         a           I*a                 I*x                                   
 |  |          ------------------ + ---------------- - ------------------             otherwise     
 |  |                  2                    ________             ________                           
 |  |                                      /      2             /      2                            
 |  |                                     /      x             /      x                             
 |  |                               2*   /   1 - --    2*a*   /   1 - --                            
 |  |                                   /         2          /         2                            
 |  \                                 \/         a         \/         a                             
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/                                                                                                   
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∫01{−a−1+x2a2+3x22a−1+x2a2+x22a(−1+x2a2)32−x42a3(−1+x2a2)32for x2∣a2∣>1ia1−x2a22+ia21−x2a2−ix22a1−x2a2otherwise dx\int\limits_{0}^{1} \begin{cases} - \frac{a}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 x^{2}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{x^{2}}{2 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{x^{4}}{2 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{x^{2}}{\left|{a^{2}}\right|} > 1 \\\frac{i a \sqrt{1 - \frac{x^{2}}{a^{2}}}}{2} + \frac{i a}{2 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{i x^{2}}{2 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx 
=
= 
  1                                                                                                 
  /                                                                                                 
 |                                                                                                  
 |  /                            2                   4                     2               2        
 |  |         a                 x                   x                   3*x               x         
 |  |- --------------- + ---------------- - ----------------- + -------------------  for ---- > 1   
 |  |        _________                3/2                 3/2             _________      | 2|       
 |  |       /       2        /      2\           /      2\               /       2       |a |       
 |  |      /       x         |     x |         3 |     x |              /       x                   
 |  |     /   -1 + --    2*a*|-1 + --|      2*a *|-1 + --|      2*a*   /   -1 + --                  
 |  |    /          2        |      2|           |      2|            /          2                  
 |  |  \/          a         \     a /           \     a /          \/          a                   
 |  |                                                                                               
 |  |                    ________                                                                   
 |  <                   /      2                                                                  dx
 |  |                  /      x                                                                     
 |  |          I*a*   /   1 - --                                                                    
 |  |                /         2                                 2                                  
 |  |              \/         a           I*a                 I*x                                   
 |  |          ------------------ + ---------------- - ------------------             otherwise     
 |  |                  2                    ________             ________                           
 |  |                                      /      2             /      2                            
 |  |                                     /      x             /      x                             
 |  |                               2*   /   1 - --    2*a*   /   1 - --                            
 |  |                                   /         2          /         2                            
 |  \                                 \/         a         \/         a                             
 |                                                                                                  
/                                                                                                   
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∫01{−a−1+x2a2+3x22a−1+x2a2+x22a(−1+x2a2)32−x42a3(−1+x2a2)32for x2∣a2∣>1ia1−x2a22+ia21−x2a2−ix22a1−x2a2otherwise dx\int\limits_{0}^{1} \begin{cases} - \frac{a}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 x^{2}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{x^{2}}{2 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{x^{4}}{2 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{x^{2}}{\left|{a^{2}}\right|} > 1 \\\frac{i a \sqrt{1 - \frac{x^{2}}{a^{2}}}}{2} + \frac{i a}{2 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{i x^{2}}{2 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx 
Integral(Piecewise((-a/sqrt(-1 + x^2/a^2) + x^2/(2*a*(-1 + x^2/a^2)^(3/2)) - x^4/(2*a^3*(-1 + x^2/a^2)^(3/2)) + 3*x^2/(2*a*sqrt(-1 + x^2/a^2)), x^2/|a^2| > 1), (i*a*sqrt(1 - x^2/a^2)/2 + i*a/(2*sqrt(1 - x^2/a^2)) - i*x^2/(2*a*sqrt(1 - x^2/a^2)), True)), (x, 0, 1))

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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